Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B

Percentage Accurate: 99.9% → 99.9%

Time: 6.4s

Alternatives: 10

Speedup: 1.0×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

2. Final simplification 100.0%

   \[\leadsto \frac{\left(x + y\right) - z}{t \cdot 2} \]

Alternative 2: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+170} \lor \neg \left(z \leq -6 \cdot 10^{+99} \lor \neg \left(z \leq -2.4 \cdot 10^{+61}\right) \land z \leq 3.7 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.2e+170)
         (not (or (<= z -6e+99) (and (not (<= z -2.4e+61)) (<= z 3.7e+105)))))
   (* (/ z t) -0.5)
   (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e+170) || !((z <= -6e+99) || (!(z <= -2.4e+61) && (z <= 3.7e+105)))) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.2d+170)) .or. (.not. (z <= (-6d+99)) .or. (.not. (z <= (-2.4d+61))) .and. (z <= 3.7d+105))) then
        tmp = (z / t) * (-0.5d0)
    else
        tmp = 0.5d0 * ((x + y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.2e+170) || !((z <= -6e+99) || (!(z <= -2.4e+61) && (z <= 3.7e+105)))) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.2e+170) or not ((z <= -6e+99) or (not (z <= -2.4e+61) and (z <= 3.7e+105))):
		tmp = (z / t) * -0.5
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.2e+170) || !((z <= -6e+99) || (!(z <= -2.4e+61) && (z <= 3.7e+105))))
		tmp = Float64(Float64(z / t) * -0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.2e+170) || ~(((z <= -6e+99) || (~((z <= -2.4e+61)) && (z <= 3.7e+105)))))
		tmp = (z / t) * -0.5;
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.2e+170], N[Not[Or[LessEqual[z, -6e+99], And[N[Not[LessEqual[z, -2.4e+61]], $MachinePrecision], LessEqual[z, 3.7e+105]]]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999996e170 or -6.00000000000000029e99 < z < -2.3999999999999999e61 or 3.69999999999999985e105 < z

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in z around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   3. Step-by-step derivation

      1. *-commutative 77.3%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

   4. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

   if -5.1999999999999996e170 < z < -6.00000000000000029e99 or -2.3999999999999999e61 < z < 3.69999999999999985e105

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in z around 0 87.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x + y}{t}} \]
   3. Step-by-step derivation

      1. +-commutative 87.2%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y + x}}{t} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y + x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+170} \lor \neg \left(z \leq -6 \cdot 10^{+99} \lor \neg \left(z \leq -2.4 \cdot 10^{+61}\right) \land z \leq 3.7 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 3: 46.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{-0.5}{t}\\ t_2 := 0.5 \cdot \frac{x}{t}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ -0.5 t))) (t_2 (* 0.5 (/ x t))))
   (if (<= y -6.5e-148)
     t_2
     (if (<= y -1.25e-212)
       t_1
       (if (<= y 1.12e-272) t_2 (if (<= y 1.45e+26) t_1 (* 0.5 (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (-0.5 / t);
	double t_2 = 0.5 * (x / t);
	double tmp;
	if (y <= -6.5e-148) {
		tmp = t_2;
	} else if (y <= -1.25e-212) {
		tmp = t_1;
	} else if (y <= 1.12e-272) {
		tmp = t_2;
	} else if (y <= 1.45e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = z * ((-0.5d0) / t)
    t_2 = 0.5d0 * (x / t)
    if (y <= (-6.5d-148)) then
        tmp = t_2
    else if (y <= (-1.25d-212)) then
        tmp = t_1
    else if (y <= 1.12d-272) then
        tmp = t_2
    else if (y <= 1.45d+26) then
        tmp = t_1
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (-0.5 / t);
	double t_2 = 0.5 * (x / t);
	double tmp;
	if (y <= -6.5e-148) {
		tmp = t_2;
	} else if (y <= -1.25e-212) {
		tmp = t_1;
	} else if (y <= 1.12e-272) {
		tmp = t_2;
	} else if (y <= 1.45e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (-0.5 / t)
	t_2 = 0.5 * (x / t)
	tmp = 0
	if y <= -6.5e-148:
		tmp = t_2
	elif y <= -1.25e-212:
		tmp = t_1
	elif y <= 1.12e-272:
		tmp = t_2
	elif y <= 1.45e+26:
		tmp = t_1
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-0.5 / t))
	t_2 = Float64(0.5 * Float64(x / t))
	tmp = 0.0
	if (y <= -6.5e-148)
		tmp = t_2;
	elseif (y <= -1.25e-212)
		tmp = t_1;
	elseif (y <= 1.12e-272)
		tmp = t_2;
	elseif (y <= 1.45e+26)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (-0.5 / t);
	t_2 = 0.5 * (x / t);
	tmp = 0.0;
	if (y <= -6.5e-148)
		tmp = t_2;
	elseif (y <= -1.25e-212)
		tmp = t_1;
	elseif (y <= 1.12e-272)
		tmp = t_2;
	elseif (y <= 1.45e+26)
		tmp = t_1;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-148], t$95$2, If[LessEqual[y, -1.25e-212], t$95$1, If[LessEqual[y, 1.12e-272], t$95$2, If[LessEqual[y, 1.45e+26], t$95$1, N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999997e-148 or -1.25000000000000011e-212 < y < 1.11999999999999994e-272

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in x around inf 41.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -6.4999999999999997e-148 < y < -1.25000000000000011e-212 or 1.11999999999999994e-272 < y < 1.45e26

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   3. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

      3. associate-*r/ 99.9%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

      4. associate-*l/ 99.7%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

      5. distribute-lft-in 99.7%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

      6. associate-+r- 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) - z\right)} \]

      7. sub-neg 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \left(\color{blue}{\left(y + x\right)} + \left(-z\right)\right) \]

      9. associate-+l+ 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(y + \left(x + \left(-z\right)\right)\right)} \]

      10. sub-neg 99.7%

          \[\leadsto \frac{0.5}{t} \cdot \left(y + \color{blue}{\left(x - z\right)}\right) \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(y + \left(x - z\right)\right)} \]

   5. Taylor expanded in z around inf 55.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 55.4%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot z}{t}} \]

      2. associate-/l* 55.3%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{t}{z}}} \]

   7. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{t}{z}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 55.3%

         \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   9. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 1.45e26 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in y around inf 66.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-212}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-272}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4: 47.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot -0.5\\ t_2 := 0.5 \cdot \frac{x}{t}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) -0.5)) (t_2 (* 0.5 (/ x t))))
   (if (<= y -5.8e-148)
     t_2
     (if (<= y -1.45e-198)
       t_1
       (if (<= y 5.5e-272) t_2 (if (<= y 7.8e+26) t_1 (* 0.5 (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = 0.5 * (x / t);
	double tmp;
	if (y <= -5.8e-148) {
		tmp = t_2;
	} else if (y <= -1.45e-198) {
		tmp = t_1;
	} else if (y <= 5.5e-272) {
		tmp = t_2;
	} else if (y <= 7.8e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z / t) * (-0.5d0)
    t_2 = 0.5d0 * (x / t)
    if (y <= (-5.8d-148)) then
        tmp = t_2
    else if (y <= (-1.45d-198)) then
        tmp = t_1
    else if (y <= 5.5d-272) then
        tmp = t_2
    else if (y <= 7.8d+26) then
        tmp = t_1
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * -0.5;
	double t_2 = 0.5 * (x / t);
	double tmp;
	if (y <= -5.8e-148) {
		tmp = t_2;
	} else if (y <= -1.45e-198) {
		tmp = t_1;
	} else if (y <= 5.5e-272) {
		tmp = t_2;
	} else if (y <= 7.8e+26) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * -0.5
	t_2 = 0.5 * (x / t)
	tmp = 0
	if y <= -5.8e-148:
		tmp = t_2
	elif y <= -1.45e-198:
		tmp = t_1
	elif y <= 5.5e-272:
		tmp = t_2
	elif y <= 7.8e+26:
		tmp = t_1
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * -0.5)
	t_2 = Float64(0.5 * Float64(x / t))
	tmp = 0.0
	if (y <= -5.8e-148)
		tmp = t_2;
	elseif (y <= -1.45e-198)
		tmp = t_1;
	elseif (y <= 5.5e-272)
		tmp = t_2;
	elseif (y <= 7.8e+26)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * -0.5;
	t_2 = 0.5 * (x / t);
	tmp = 0.0;
	if (y <= -5.8e-148)
		tmp = t_2;
	elseif (y <= -1.45e-198)
		tmp = t_1;
	elseif (y <= 5.5e-272)
		tmp = t_2;
	elseif (y <= 7.8e+26)
		tmp = t_1;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e-148], t$95$2, If[LessEqual[y, -1.45e-198], t$95$1, If[LessEqual[y, 5.5e-272], t$95$2, If[LessEqual[y, 7.8e+26], t$95$1, N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.7999999999999997e-148 or -1.45e-198 < y < 5.4999999999999999e-272

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in x around inf 40.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -5.7999999999999997e-148 < y < -1.45e-198 or 5.4999999999999999e-272 < y < 7.8e26

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in z around inf 55.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   3. Step-by-step derivation

      1. *-commutative 55.1%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

   4. Simplified 55.1%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

   if 7.8e26 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in y around inf 66.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-272}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 5: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.95e+32) (* 0.5 (/ (- x z) t)) (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.95e+32) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.95d+32) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * ((x + y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.95e+32) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.95e+32:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.95e+32)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.95e+32)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.95e+32], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.95e32

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in y around 0 82.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x - z}{t}} \]

   if 1.95e32 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in z around 0 86.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x + y}{t}} \]
   3. Step-by-step derivation

      1. +-commutative 86.4%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y + x}}{t} \]

   4. Simplified 86.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y + x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]

Alternative 6: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1060000000:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1060000000.0) (* 0.5 (/ (- x z) t)) (* 0.5 (/ (- y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1060000000.0) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1060000000.0d0) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * ((y - z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1060000000.0) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1060000000.0:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((y - z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1060000000.0)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(Float64(y - z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1060000000.0)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((y - z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1060000000.0], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.06e9

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in y around 0 83.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x - z}{t}} \]

   if 1.06e9 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in x around 0 81.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1060000000:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 7: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + \left(x - z\right)\right) \cdot \frac{0.5}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (+ y (- x z)) (/ 0.5 t)))

C

double code(double x, double y, double z, double t) {
	return (y + (x - z)) * (0.5 / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y + (x - z)) * (0.5d0 / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y + (x - z)) * (0.5 / t);
}

Python

def code(x, y, z, t):
	return (y + (x - z)) * (0.5 / t)

Julia

function code(x, y, z, t)
	return Float64(Float64(y + Float64(x - z)) * Float64(0.5 / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y + (x - z)) * (0.5 / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y + N[(x - z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

2. Taylor expanded in x around 0 98.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
3. Step-by-step derivation

   1. associate-*r/ 98.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

   2. associate-*l/ 97.9%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

   3. associate-*r/ 97.9%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

   4. associate-*l/ 97.7%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

   5. distribute-lft-in 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   6. associate-+r- 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) - z\right)} \]

   7. sub-neg 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)} \]

   8. +-commutative 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \left(\color{blue}{\left(y + x\right)} + \left(-z\right)\right) \]

   9. associate-+l+ 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(y + \left(x + \left(-z\right)\right)\right)} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{t} \cdot \left(y + \color{blue}{\left(x - z\right)}\right) \]

4. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(y + \left(x - z\right)\right)} \]

5. Final simplification 99.6%

   \[\leadsto \left(y + \left(x - z\right)\right) \cdot \frac{0.5}{t} \]

Alternative 8: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{t}{y + \left(x - z\right)}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 0.5 (/ t (+ y (- x z)))))

C

double code(double x, double y, double z, double t) {
	return 0.5 / (t / (y + (x - z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 / (t / (y + (x - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 / (t / (y + (x - z)));
}

Python

def code(x, y, z, t):
	return 0.5 / (t / (y + (x - z)))

Julia

function code(x, y, z, t)
	return Float64(0.5 / Float64(t / Float64(y + Float64(x - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 / (t / (y + (x - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 / N[(t / N[(y + N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

2. Taylor expanded in x around 0 98.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
3. Step-by-step derivation

   1. associate-*r/ 98.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

   2. associate-*l/ 97.9%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

   3. associate-*r/ 97.9%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

   4. associate-*l/ 97.7%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

   5. distribute-lft-in 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   6. associate-+r- 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) - z\right)} \]

   7. sub-neg 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(x + y\right) + \left(-z\right)\right)} \]

   8. +-commutative 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \left(\color{blue}{\left(y + x\right)} + \left(-z\right)\right) \]

   9. associate-+l+ 99.6%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(y + \left(x + \left(-z\right)\right)\right)} \]

   10. sub-neg 99.6%

       \[\leadsto \frac{0.5}{t} \cdot \left(y + \color{blue}{\left(x - z\right)}\right) \]

4. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(y + \left(x - z\right)\right)} \]
5. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(y + \left(x - z\right)\right)}{t}} \]

   2. associate-/l* 99.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{y + \left(x - z\right)}}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{y + \left(x - z\right)}}} \]

7. Final simplification 99.8%

   \[\leadsto \frac{0.5}{\frac{t}{y + \left(x - z\right)}} \]

Alternative 9: 46.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.2e+23) (* 0.5 (/ x t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+23) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.2d+23) then
        tmp = 0.5d0 * (x / t)
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.2e+23) {
		tmp = 0.5 * (x / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.2e+23:
		tmp = 0.5 * (x / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.2e+23)
		tmp = Float64(0.5 * Float64(x / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.2e+23)
		tmp = 0.5 * (x / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.2e+23], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.20000000000000008e23

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in x around inf 42.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if 2.20000000000000008e23 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

   2. Taylor expanded in y around inf 65.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 10: 37.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 0.5 (/ x t)))

C

double code(double x, double y, double z, double t) {
	return 0.5 * (x / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 * (x / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 * (x / t);
}

Python

def code(x, y, z, t):
	return 0.5 * (x / t)

Julia

function code(x, y, z, t)
	return Float64(0.5 * Float64(x / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 * (x / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]

2. Taylor expanded in x around inf 36.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

3. Final simplification 36.5%

   \[\leadsto 0.5 \cdot \frac{x}{t} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))